Geometry 12.2 – 12.5 Prep for the STAR Test

Geometry 12.2 Pyramids

The lateral area of a regular pyramid with n lateral faces is n times the area of one lateral face ) Find the lateral area and total area of this regular pyramid. Let’s solve #7 as we learn the formula for lateral area!! The best way to know the formulas is to understand them rather than memorize them. L.A. = nF 6 10 A = ½ b(h) A = 3(10) A = 30 LA = nF LA = 6(30) LA = 180 square units OR… The lateral area of a regular pyramid equals half the perimeter of the base times the slant height. L.A. = ½ pl Why? Imagine a curtain on the ground around the pyramid(perimeter) and you are pulling the curtain up the slant height. Since the curtain will wrap around triangles we need the ½. LA = ½ pl LA = ½ 36(10) LA = 180 square units We have 6 triangles!

7) Find the lateral area and total area of this regular pyramid. The total area of a pyramid is its lateral area plus the area of its base T.A. = L.A. + B That makes sense! A = ½ a(p) √3 A = ½ 3√3(36) A = 3√3(18) A = 54√3 TA = LA + B TA = √3 sq. units

Let’s solve #9 as we learn the formula volume!! 9. Find the volume of a regular square pyramid with base edge 24 and lateral edge 24. Draw a square pyramid with the given dimensions. 24 The volume of a pyramid equals one third the area of the base times the height of the pyramid. Why? Because a rectangular prism with the same base and same height would be V = bh and the pyramid, believe it or not, would hold 1/3 the amount of water. h l 1/3 the volume of its prism. V = 1/3 B(h) V = 1/3 24(24)(h) V = 8(24)(h) V = 192(h) Must be a √ x 2 = (12√3) 2 12√2 V = 192(12√2) V = 2304√2 sq. units

12.3 Cylinders and Cones

To find lateral area (L.A.): Multiply circumference by height circumference height Take a can of soup Peel off the label

To find total area (T.A.):Take the label (LA) Pop the top and find the area of the base TopBottom Add 2 base areas to the LA A = πr²

To find volume (V):Start with the area of the base Multiply it by height H r That’s how much soup is in the can ! A = πr²

Lateral Area of a Cylinder: L.A. The lateral area of a cylinder equals the circumference of a base times the height of the cylinder. L.A = 2πrH LA = 12π 8 = 96π square units L.A = dπH L.A = CH which is 6 8

Total Area of a Cylinder: T.A. The total area of a cylinder is the lateral area plus twice the area of a base. T.A = L.A. + 2B TA = 96π + 2(π 6²) = 96π + 2(36π) = 96π + 72π = 168π square units 6 8 T.A. = 2πrH + 2πr² which is

Volume of a Cylinder: V The volume of a cylinder equals the area of a base times the height of the cylinder. V = πr²H V = (π 6²) 8 = 36π 8 = 288π cubic units 6 8

Lateral Area of a Cone: L.A. LA = π 6 10 = 60π square units L.A = πrl

Total Area of a Cone: T.A. T.A = L.A. + B TA = 60π + (π 6²) = 60π + 36π = 96π square units T.A. = πrl + πr² which is

Volume of a Cone: V The volume of a cone equals one third the area of the base times the height of the cone. V = πr²h V = 1/3 π 6² 8 = 96π cubic units

12.4 Spheres

Surface Area Formula Surface Area = r What does this represent? It represents the area of a circle with radius = r.

Volume Formula Volume = A way to remember which formula is cubed… The volume formula has two 3’s and volume is measured in units 3 !!

1. radius = 10 Area: _____ Volume: _____ Surface Area = = 4π(10) 2 = 4 π (100) = 400π units 2 Volume =

Geometry 12.5 Areas and Volumes of Similar Solids

If the scale factor of two solids is a:b, then (1)the ratio of corresponding perimeters is a:b (2)the ratio of base areas, of lateral areas, and of the total area is a²:b² (3) the ratio of volumes is a³:b³ Scale Factor SCALE FACTOR: 1:2 Base circumference: 6π:12π 1:2 Lateral areas: 15π:60π 1:4 Volumes: 12π:96π 1:

Exercises Find the missing information scale factor2 : 5 ratio of base perimeters ratio of heights1 : 3 ratio of lateral areas4 : 49 ratio of total areas ratio of volumes125 : : :5 4:25 8:125 1:9 1:27 2:7 8:343 5:6 25:36 3:10 9:100

HW Midpoint WS